What does a saddle point indicate in two-person games?

A saddle point in the context of two-person games reflects a situation where both players are employing optimal pure strategies, resulting in a stable outcome for the game. It is the point at which the chosen strategies for both players lead to the best possible outcomes given the strategies of the opponent.

In more detail, at a saddle point, the strategy for one player represents the best response against the strategy of the other player, ensuring that neither player can improve their payoff by unilaterally changing their strategy. This stability signifies that both players have chosen their optimal approaches, leading to a scenario where they cannot benefit from deviating from these strategies.

This is crucial in game theory as it helps identify equilibrium strategies, guiding players to achieve the most favorable outcomes without necessitating mixed strategies or randomization, which would otherwise introduce uncertainty into their decision-making process.